>> How can you advertise a game on the App Store? Does Apple offer some internal method of listing and ranking games to influence downloads?

You become a top app or trending app for the first question; you pay Apple with their fairly easy to find advertising system for the second question.

>> Also, how do most iOS game devs monetize and get the word out about their games?

Every game is different. Be careful about "most games", since most games end up as commercial failures. What may be a good fit for one game may be a bad fit for another. For monitization, it is currently common to have a freely available game. Maybe your game would be best purchased outright for a few cents, or perhaps your game would be better if it were free for the first two levels, or perhaps free with in-game purchases, or entirely ad supported. As for getting the word out, that is advertising; advertising for an ultra-violent FPS is very different than advertising for a child's game. Again with advertising, every game is different, and there is no common formula for success.

Like frob said, advertising. However, also consider using something
like TapJoy to generate installs and move your game up in the download
charts which will then help you generate more organic installs.

All of this requires money, though. For free, try submitting your game to
bloggers or review sites in order for them to evaluate it and possibly
post it on their sites.